Chen, Linan; Stroock, Daniel W. The fundamental solution to the Wright-Fisher equation. (English) Zbl 1221.35013 SIAM J. Math. Anal. 42, No. 2, 539-567 (2010). This paper is devoted to the study of the fundamental solution to the Cauchy initial value problem for the Wright-Fisher equation \[ \partial_t u(x,t) = x(1-x)\partial^2_x u(x,t),\quad (x,t)\in (0,1)\times(0,\infty), \]with boundary conditions \[ \lim_{t\rightarrow 0} u(x,t) = \varphi(x),\quad x\in(0,1);\quad u(0,t) = u(1,t) = 0,\quad t\in(0,\infty). \]The authors use a stochastic approach and associate the Itô stochastic differential equation to the Wright-Fisher equation in the form \[ X(t,x) = x+\int_0^t\sqrt{2X(\tau,x)(1-X(\tau,x))}\,dB(\tau), \]proving that the fundamental solution of the Wright-Fisher equation is the density for the distribution of the solution to the Itô equation. Further results involve estimates on the fundamental solution to the Wright-Fisher equation as \(t\rightarrow 0\) and Taylor series expansions of the solutions. Reviewer: Dora Selesi (Novi Sad) Cited in 1 ReviewCited in 10 Documents MSC: 35A08 Fundamental solutions to PDEs 35B45 A priori estimates in context of PDEs 35K65 Degenerate parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) Keywords:Itô stochastic differential equation PDFBibTeX XMLCite \textit{L. Chen} and \textit{D. W. Stroock}, SIAM J. Math. Anal. 42, No. 2, 539--567 (2010; Zbl 1221.35013) Full Text: DOI